Computer Simulation in Physics and Engineering

Martin Oliver Steinhauser, Fraunhofer-Institute for High-Speed Dynamics, Ernst-Mach-Institute, EMI, Freiburg, Germany.

This work is a needed reference for widely used techniques and methods of computer simulation in physics and other disciplines, such as materials science. Molecular dynamics computes a molecule's reactions and dynamics based on physical models; Monte Carlo uses random numbers to image a system's behaviour when there are different possible outcomes with related probabilities. The work conveys both the theoretical foundations as well as applications and "tricks of the trade", that often are scattered across various papers. Thus it will meet a need and fill a gap for every scientist who needs computer simulations for his/her task at hand. In addition to being a reference, case studies and exercises for use as course reading are included.

Preface

1. Introduction to Computer Simulation
1.1 Historical Background
1.2 Theory, Modeling and Simulation in Physics
1.3 Reductionism in Physics
1.4 Basics of Ordinary and Partial Differential Equations in Physics
1.5 Numerical Solution of Differential Equations: Mesh-Based vs. Particle Methods
1.6 The Role of Algorithms in Scientific Computing
1.7 Remarks on Software Design
1.8 Summary

2. Fundamentals of Statistical Physics
2.1 Introduction
2.2 Elementary Statistics
2.3 Introduction to Classical Statistical Mechanics
2.4 Introduction to Thermodynamics
2.5 Summary

3. Inter- and Intramolecular Short-Range Potentials
3.1 Introduction
3.2 Quantum Mechanical Basis of Intermolecular Interactions
3.2.1 Perturbation Theory
3.3 Classical Theories of Intermolecular Interactions
3.4 Potential Functions
3.5 Molecular Systems
3.6 Summary

4. Molecular Dynamics Simulation
4.1 Introduction
4.2 Basic Ideas of MD
4.3 Algorithms for Calculating Trajectories
4.4 Link between MD and Quantum Mechanics
4.5 Basic MD Algorithm: Implementation Details
4.6 Boundary Conditions
4.7 The Cutoff Radius for Short-Range Potentials
4.8 Neighbor Lists: The Linked-Cell Algorithm
4.9 The Method of Ghost Particles
4.10 Implementation Details of the Ghost Particle Method
4.11 Making Measurements
4.12 Ensembles and Thermostats
4.13 Case Study: Impact of Two Different Bodies
4.14 Case Study: Rayleigh-Taylor Instability
4.15 Case Study: Liquid-Solid Phase Transition of Argon

5. Advanced MD Simulation
5.1 Introduction
5.2 Parallelization
5.3 More Complex Potentials and Molecules
5.4 Many Body Potentials
5.5 Coarse Grained MD for Mesoscopic Systems

6. Outlook on Monte Carlo Simulations
6.1 Introduction
6.2 The Metropolis Monte-Carlo Method
6.2.1 Calculation of Volumina and Surfaces
6.2.2 Percolation Theory
6.3 Basic MC Algorithm: Implementation Details
6.3.1 Case Study: The 2D Ising Magnet
6.3.2 Trial Moves and Pivot Moves
6.3.3 Case Study: Combined MD and MC for Equilibrating a Gaussian Chain
6.3.4 Case Study: MC of Hard Disks
6.3.5 Case Study: MC of Hard Disk Dumbbells in 2D
6.3.6 Case Study: Equation of State for the Lennard-Jones Fluid
6.4 Rosenbluth and Rosenbluth Method
6.5 Bond Fluctuation Model
6.6 Monte Carlo Simulations in Different Ensembles
6.7 Random Numbers Are Hard to Find

7. Applications from Soft Matter and Shock Wave Physics
7.1 Biomembranes
7.2 Scaling Properties of Polymers
7.3 Polymer Melts
7.4 Polymer Networks as a Model for the Cytoskeleton of Cells
7.5 Shock Wave Impact in Brittle Solids

8. Concluding Remarks

A Appendix
A.1 Quantum Statistics of Ideal Gases
A.2 Maxwell-Boltzmann, Bose-Einstein- and Fermi-Dirac Statistics
A.3 Stirling's Formula
A.4 Useful Integrals in Statistical Physics
A.3 Useful Conventions for Implementing Simulation Programs
A.4 Quicksort and Heapsort Algorithms
A.4 Selected Solutions to Exercises
Abbreviations
Bibliography
Index